Transient heat flow

Complicated problems of transient heat flow can be resolved by computer. Typical time-temperature curves for non-steady cooling are shown in Figures 16.1 and 16.2, and the subject is met again in Section 26.2. 

TRANSIENT HEAT FLOW IN A SEMI-INFINITE SOLID... 

Fig. 4-3 Nomenclature lor transient heat flow in a semi- infinite solid...

J. H. Blackwell, Transient Heat Flow from a Thin Circular Disk Small-Time Solution, Journal of the Australian Mathematical Society, Vol. XIV, pp. 433-442,1972. 

N. R. Keltner, Transient Heat Flow in Half-Space Due to an Isothermal Disk on the Surface, Journal of Heat Transfer (95) 412-414,1973. 

E. J. Normington and J. H. Blackwell, Transient Heat Flow From Constant Temperature Spheroids and the Thin Circular Disk, Quarterly Journal of Mechanics and Applied Mathematics Vol. 17, pp. 65-72,1964. 

The parameter a was called the thermal diffusivity by Kelvin and the thermometric conductivity by Ma.xwell. but Kelvin s expression has been generally adopted. It measures the change in temperature that would be produced in unit volume of the substance by the quantity of heal that flows in unit time across unit area of a layer of the sub.stance of unit thickness with unit temperature difference between its faces [3]. It is the parameter that determines the non-steady state temperature distribution in the absence of heat generation and convection and is therefore essential for transient heat flow calculations. 

To measure diffusivity it is only necessary to know the change in temperature with time at three collinear points in the direction of heat flow. For conductivity on the other hand, although the temperature gradient can easily be obtained, the heat flu.x cannot be measured at a point the total power to the heat source is known, but it is difficult to control or calculate the movement of the energy. Thus diffusivity is easier to measure but it has received less attention, although it is the essential parameter for transient heat flow calculations. The scatter in the diffusivity data in the literature is not as great as for conductivity (possibly because not as many people have tried to mea.surc it ). 

Transient conduction conditions occur in polymer processing. Appendix A derives Eq. (A.14) for one-dimensional transient heat flow, which contains the thermal diffusivity a. This is the combination k/pCp of the thermal conductivity k, density p and specific heat Cp. For most polymer melts a is approximately equal to O.lmm s" (Fig. 5.3). For the melting of low-density polyethylene in an extruder, typical conditions are a barrel temperature of To = 220 °C, an initial polymer temperature Tp = 20 °C, and a melting process complete at T = 120 °C. Consequently, using Eq. (C.19), after a contact time t, the melt front is at a distance from the barrel given by... 

In Appendix A, Eq. (A. 14) is the differential equation for onedimensional transient heat flow. Many finite element packages allow the... 

Hopper, F.C. and Lepper, F.R., Transient heat flow apparatus for the determination of thermal conductivities, ASHVE Trans., 56 309-324 (1950). 

For other than flat plates, heat flux lines are seldom parallel, rarely steady. In transient heat flow, determination of the temperature at a given time and point within the load necessitates use of the finite element method. 

For a more rapid method see hot-wire method for rapid, transient heat flow methods see thermal diffusivity. Thermal Ceding Test Refractories subject to repeated changes in temperature (thermal cycling) lose strength. ASTM... 

Liebmann G (1955) The solution of transient heat flow and heat transfer problems by relaxation. Brit J Appl Phys 6 129-135... 

When the temperature changes with time at any point in a material, the rate of change of temperafure wifh time is dependent on thermal diffusivity. Thermal diffusivity is the ratio of thermal conductivity to heat capacity per unit volume. The heat flux for a transient heat flow can then be written as Equation 16.10. 

In case of steady state heat conduction, the material property is the conductivity which can be calculated once the heat loss from the body is known and the boundary temperature is measured. In case of transient heat flow, the main factor is the diffusivity a which is equal to the ratio of the conductivity and the heat content of the body. Hence equation 10.8 is the basic governing equation of transient heat transfer with boundary conditions relevant for textile materials. Transient state heat conduction is related to instantaneous conduction of heat fi-om the surface of the body to the clothing. Instantaneous heat transfer can be related to the warmth or coolness to touch and the warm-cool feeling of any clothing can be quantified. 

Transient-heat-flow Methods. The axial and radial heat-flow methods discussed above depend on attaining a steady-state condition within the test specimen. This boundary condition is readily understood theoretically and easy to test experimentally. A disadvantage lies in the long, sometimes mctremely long, times required to achieve the condition. Under certain circumstances thermal conductivity can he determined by transient methods, where test times are much shorter. However, boundary conditions are difficult to analyze theoretically and often very approximate models are used for calculational purposes. 

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